1. I encountered a mathematical symbol that I don't know the meaning of and I'm curious about it... It's a large(r) circle inscribed with a "plus sign." Can anyone at least give me a name for it so I can look it up to read more about it, if not explain its usage to me? If you want to see it, go to the Spinor article on Wikipedia and then down to the "Even Dimensions" section.

2.

3. Looks like it means direct sum.
http://en.wikipedia.org/wiki/%E2%8A%95

I'm a math noob, but all I did was search for a list of Latex symbols, find the circle with cross and copy/paste it to wikipedia search.

4. The ⊕ symbol is most often used to mean "direct sum". The direct sum is a construction which, in many contexts, looks a lot like the cartesian product except it carries with it more structure. For example, if V and W are vector spaces, we let V⊕W be the vector space consisting of pairs (v,w), v and w vectors in V and W, respectively. The extra bits of structure that set this apart from the cartesian product are addition:

(v,w)+(v',w') = (v+v',w+w')

and scalar multiplication:

k(v,w) = (kv,kw)

In some sense, the direct sum is the simplest way of creating a vector space which allows you to "add two vectors from different spaces".

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